Prediction of properties of boron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-icosahedral nanosheet by bond-addictive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {M}}$$\end{document}M-polynomial

Nanosheets with boron elements have excellent characteristics which makes the boron polymorphs unique and super hard. A boron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-icosahedral nanosheet in crystalline form has superconductivity and thermal electronic properties. In theoretical chemistry and QSPR/QSAR study, a topological descriptor is an important analytical tool. It helps to analyse the structure and its properties and also correlates the with numerical expressions. The valence-based M-polynomial provides quantitative measures of molecular properties based on their geometric, electrostatic, and quantum chemical characteristics. In this article, the QSPR/QSAR analysis is performed for this nanosheet and the analytical expressions are validated with original synthesized data, and received excellent correlation values of 0.9835 and 0.9932. The mathematical expression of the structure is analysed and the indices are compared graphically and numerically.

Boron is an interesting and complex element, many aspects of which are still to be explored.The properties of boron are found between metals and insulators.While boron has only three valence electrons, which would favor metallicity, they are localized enough to produce insulating states.However, pressure, temperature, and impurities can easily shift this subtle balance between metallic and insulating states.Pure boron is one of the best alternatives to carbon fullerenes (CFs) and nanotubes (CNTs), which exhibit superior properties, in the form of novel solids and nanostructures, such as quasiplanar clusters, quasi-crystals, nanosheets, nanoribbons, nano chains, and nanotubes 1 .Besides being the only non-metal element in Group III, boron is unique in its structural complexity and has exceptional chemical and physical properties, including low densities, high melting points, and high hardness 2 .Initially, Boron exists in three crystalline forms, α − B 12 , β − B 106 and γ − B 28 3 .Later different forms of boron crystalline have been synthesized, such as α -rhombohedral, β-rhombohedral, tetragonal, γ-orthorhombic, and α-Ga type.In addition, there are amorphous phases and nanosized structures 4 .One of them is the B 12 icosahedral that is linked together by "inter-icosahedral covalent units" or "chains" 5 .The boron-rich ceramics based on icosahedral are second only to diamonds as hard materials.When compared to diamond-based materials, this class of ceramics offer low density, better thermal and chemical resistance, and ease of mass production.
In boron α-icosahedral nanosheet, each crystal contains an icosahedron molecule of B 12 , which is linked to form a three-dimensional network 1 as shown in Fig. 1.A regular icosahedron has 12 vertices, 30 edges and 20 faces.The icosahedral boron sheet, B 12 and B 20 have been proposed in recent years with special properties.Kah et al. 6 proposed many icosahedral nanosheets based on B 12 clusters, and Zhou et al. 7 presented an antifer- romagnetic metallic B 20 sheet.Higashi et al. 8 investigated the first 2D icosahedral B 12 networks.The icosahedral nanosheet bonding is complex and was well explained by Emin 9 .The boron allotropes attract major material researchers since they exhibit properties like thermal conductivity, hardness, and neutron scattering length 10 .The novel icosahedral structures exhibit interesting chemical bonding and electronic properties and are structurally and energetically stable.Additionally, these α-icosahedral nanosheets, which are a gapless system, exhibit semiconducting properties, suggesting an application in nanoelectronics and computer chips.is a good choice, In industrial semiconductor applications like solar cells with high solar light conversion efficiency, the icosahedal boron nanosheet is a prominent component 11 .
In a molecular graph, each edge of a molecule corresponds to a chemical bond between atoms, while each vertex and degree denotes an atom and valence of the atom.In order to characterize the structural features of these molecules, several theoretical tools are employed.A topological index can be used to model relationships between chemical structures and their corresponding biochemical and physicochemical activities 12,13 .Large combinatorial chemical libraries are required to compute the physicochemical properties of a structure.These include novel development methods such as topological structural descriptors, combinatorial quantum chemistry tools for functional group analysis, shape-activity relations, and topological attributes of electron densities, etc.The degree-based topological indexes are used extensively in network science for investigating networks, in which the indexes are calculated based on the degrees of the graph.A breakthrough was made in degree-based indices by Deutsch and Klavzar 14 , introducing the M-polynomial.Readers can refer to [15][16][17][18] for recent work in M-polynomial and topological indices.
Boron α-icosahedral nanosheets are grabbing immense attention due to their numerous applications in emerging technologies.Thus, understanding the properties of these structures is imperative for industrial applications.In this paper, the degree-based structure analysis of α-icosahedral nanosheet is performed using M-polynomial.The analytical expressions for some prominent indices are evaluated and their graphical representations are plotted using the numerical values of these indices and compared.The shear modulus and Young's modulus of the icosahedral nanosheet are compared against its structural parameters, which helps to predict the properties of numerous additional boron allotropes.

Computational techniques
A chemical compound can be modeled as a simple graph, χ with vertex and edge sets, V(χ) and E(χ) respectively.The valency of an atom is denoted by d µ of the vertex µ ∈ V(χ) , whereas the maximum degree over all the ver- tices of χ is denoted by .The degree of the vertex of boron α-icosahedral nanosheet is illustrated in Fig. 2. The set are consider, The M-polynomial 14 for simple connected graph, χ is defined by where m kh (χ) be the total number of edges µη ∈ E(χ) such that {d µ , d η } = {k, h} .The bond additive is the function from χ into R specified as real numbers β k,h , (k, h) ∈ D induced by β(χ) = (k,h)∈D d kh β kh .The degree-based structural descriptors for χ , where f(d µ , d η ) is the function of degree based indices is depicted as A brief discussion of bond additive degree-based indices is given below regarding the above-specified real numbers, β kh .First degree-based structure descriptors were studied 19 and developed 20 with the Zagreb index, M 1 (χ) defined by β kh = k + h based on the square root of the vertex degrees to analyze the influence of total electron energy on structure.The next analogous of Zagreb index is second Zagreb index, M 2 (χ) represented as β kh = kh .These indices help in analyzing the complexity of the molecular system and increase with extent branching of the carbon skeleton.The other analogous of Zagreb index are augumented Zagreb index , AZ(χ) 21 and hyper Zagreb index, HM(χ) 22 is defined by β kh = kh k+h−2 3 and are used to analyze new drugs' molecular structures and to understand their biological and chemical properties.Based on the inverse value of vertex degree, the other invariant of Zagreb index, modified Zagreb, M m 2 (χ) 23 defined by β kh = 1 kh is evolved.Several studies have demonstrated that the augmented Zagreb index can predict the temperature at which octanes and heptanes form.These variants of Zagreb indices can be used for determining the isomerism of ZE, chirality, heat formation, and heterogeneity of hetero systems.
Based on the degrees of the end vertices of χ , several methods have been proposed to examine the branching properties of alkanes.In 1975 Milan Randić 24 developed the topological index of graph, χ under the label "molecular connectivity index" in the description R −1 and R −1/2 .A general Randić index, R d (χ) latterly developed by Bollobas and Erdos 25 by substituting R −1 and R −1/2 with a real integer d is defined as β kh = (kh) d .The other variant of randić index are reciprocal randić, RR d (χ) 26 and harmonic index, H(χ) 27 are represented as k+h .Graph eigenvalues were analyzed by Favaron et al. 28 in relation to harmonic indices.A correlation has been demonstrated between these variants of randic index and various physicochemical properties of alkanes, including the formation of enthalpies, surface areas, vapor pressure, boiling points, Kovats constants, and so on 29 .
Symmetric division degree index, SSD(χ) 30 is a great tool for predicting polychlorobiphenyl surfaces is defined by 31 , which greatly enhances the physicochemical predic- tion of the First Zagreb index, and it is defined as β kh = k 2 + h 2 .An important tool for estimating octane isomer surface area is the inverse sum index I(χ) 32 defined as β kh = kh k+h .And sigma index, σ (χ) is given by β kh = (k − h) 2 .By analyzing the above discussion, it is evident that the bond additive degree is a significant aspect to investigate the physicochemical properties of molecular structures.Table 1 outlines the formulations for the M-polynomial method.
The operators are required which relate the degree-based topological descriptors with the M-polynomial, Table 1.The derivation of vertex-degree M-polynomials.

Significance of molecular descriptors
In quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), the topological descriptors, ' describes' the molecular structure's properties or activities in mathematical terminologies.QSAR/QSPR mathematically correlates the physicochemical properties or biological activity of chemical compounds with molecular descriptors.The base for this idea, QSAR/QSPR modelling is many chemical compounds have been implicitly equated with the overall risks which cause acute effects on human health.Some of the pesticide compounds are highly toxic and few may cause cancer.
The toxicological testing of an active ingredient is usually limited.To estimate and rank the potentially hazardous chemicals, it is essential to develop an accurate and simple method 33 .Thus, it is a critical need to analyze and understand the structural properties of molecular compounds.Linear regression, multiple linear regression, logistic regression 34 , efficient linear method 35 , principal component analysis 36 , partial least square regression 37 , decision tree 38 and random forest 39 are the modelling techniques or methods that are used to analyze or predict the molecular compounds.In our study, linear regression method is deployed for statistical analysis of boron α -icosahedral nanosheet.The graphica flowchart insisting on the topological descriptors and their potential uses is exhibited in Fig. 8.

Variant of boron sheets and its descriptors
Boron has recently received a lot of attention due to its diverse chemical properties and similarities to carbon.Due to the large number of allotropes and complex bonding nature of boron, many are interested to study its crystal structures and stability 40 42 , β 12 -borphene nanosheet 43 were analyzed through regression analysis.The above-mentioned boron sheet is illustrated in Fig. 9.The degree-vertex value of the base structure of boron sheets is listed in Table 5.
In our study, we investigate the elastic, geometric, thermodynamic, and mechanical properties of the boron sheets.An elastic constant is used to determine the mechanical properties of a material and describe its ability to resist deformation by external forces.With elastic constant, some mechanical properties such as Young's modulus E, bulk modulus B and Shear modulus G can be determined.The elastic properties are closely related to the thermodynamic properties like melting point, heat capacity, vacancy defect, and temperature.The Young's modulus, E, and Shear modulus, G data of various boron nanosheets are summarized in Table 6 [44][45][46] .The Young's modulus (N/m) indicates a material's ability to withstand changes in length when brought under tension or compression and shear modulus (GPa) is a measure of elastic shear material's stiffness that reflects body rigidity.

Properties analysis and theoretical prediction
The mechanical properties, Young's modulus and shear modulus of the above-mentioned boron sheets are analyzed with topological descriptors by a regression model.Legendre 47 and Gauss 48 introduced the least squares approach to linear regression in 1805 and 1809 respectively.Regression analysis is a statistical technique that determines the correlation between two or more variables.The correlation coefficient ranges from 1 to -1.The perfect positive and negative correlation is 1 and -1 where near 0 indicates weak correlation.A correlation coefficient and regression analysis are used to derive the equation connecting the descriptors and properties.The linear regression model, where M is the mechanical properties of the boron nanosheets, and TD is topological descriptors.Using SPSS software 49,50 , the invariant, i and regression coefficient, j can be calculated.The correlation coefficients between M = i + j(TD)    where E is Young's modulus.The molecular characteristics with a greater dimension can be predicted with an appropriate regression model.In Fig. 10, the scatter plots for the highest correlated properties and descriptors are shown.

y 2
Figure4shows the graphical illustration of the M-polynomial function of I α (s, p)|s = 4 and p = 5 .Thus, M(I α (s, p); y, z) can be formulated as,
Using a degree-vertex M-polynomial graph technique, the topological indices of the boron α-icosahedral nanosheet are determined.The structural characterization is used to analyse the topological properties of boron α-icosahedral nanosheet, by combining quantum chemical descriptors with nanosheet results.This research could provide a crucial tool for determining the significance of nanosheets in many areas, such as material science, drug discovery, and predictive toxicology.Furthermore, the topological indices are used in the study of boron α-icosahedral nanosheets and provide QSAR expressions that predict several molecular proper- ties such as band gap, optical and electronic stability, molecular density, enthalpies, conductivity, and so on.In this research, we correlate our theoretical results with the shear modulus and Young's modulus original data synthesized in recent years, which showed a high correlation of 0.9835 and 0.9932 with hyper Zagreb.This type of research has not been explored earlier.So, it has a significant contribution to research by finding a correlation between topological indices and properties of boron allotropes.This allows us to explore other nanosheets, it is left as an open problem for future research.E = 76.659(HM)− 2898.5

Table 5 .
Experimental data for Young's modulus and shear modulus of boron nanosheets.

Table 7 .
51,52ecent work on QSPR analysis by linear regression method, readers can refer51,52.The correlation table indicates that these boron derivatives have strong correlations within themselves for both chemical attributes.In comparison to other indices, the hyper Zagreb index has a strong correlation for Young's modulus and shear modulus.The linear regression model for shear modulus is shown below, where G is shear modulus and HM is hyper Zagreb index.Similarly, the linear regression equation for Young's modulus is determined as follows G = 165.46(HM)− 1252.8

Table 7 .
Correlation coefficient between properties and descriptors.